Axiom ax-mulass 7877

Description: Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7835. Proofs should normally use mulass 7905 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulass	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Detailed syntax breakdown of Axiom ax-mulass

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7772	. . . 4 class ℂ
3	1, 2	wcel 2141	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2141	. . 3 wff 𝐵 ∈ ℂ
6		cC	. . . 4 class 𝐶
7	6, 2	wcel 2141	. . 3 wff 𝐶 ∈ ℂ
8	3, 5, 7	w3a 973	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ)
9		cmul 7779	. . . . 5 class ·
10	1, 4, 9	co 5853	. . . 4 class (𝐴 · 𝐵)
11	10, 6, 9	co 5853	. . 3 class ((𝐴 · 𝐵) · 𝐶)
12	4, 6, 9	co 5853	. . . 4 class (𝐵 · 𝐶)
13	1, 12, 9	co 5853	. . 3 class (𝐴 · (𝐵 · 𝐶))
14	11, 13	wceq 1348	. 2 wff ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶))
15	8, 14	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

This axiom is referenced by:  mulass  7905